Computational Methods in Sciences and Engineering 2003: by T. E. Simos

By T. E. Simos

Because the Nineteen Seventies, many major insights were won into numerous components of computational equipment in sciences and engineering. New difficulties and methodologies have seemed in a few parts of sciences and engineering. there's continually a necessity in those fields for the development of knowledge alternate. the purpose of this ebook is to facilitate the sharing of principles, difficulties and methodologies among computational scientists and engineers in numerous disciplines. prolonged abstracts of papers at the advances concerning computational tools in sciences and engineering are supplied. The ebook in short describes new tools in numerical research, computational arithmetic, computational and theoretical physics, computational and theoretical chemistry, computational biology, computational mechanics, computational engineering, computational medication and excessive functionality computing.

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Basically it has been investigated boundaryvalue problems which consist of ordinary differential equations with continuous coefficients and end-point boundary conditions. The purpose of this work is to extend some fundamental spectral properties of regular Sturm-Liouville problems to special kind discontinuous boundary value problem, which consist of a Sturm-Liouville equation together with eigendependent boundary and transmission conditions. Namely we shall consider one discontinuous eigenvalue problem which consist of Sturm-Liouville equation.

Thus, for the first one we propose the numerical method of characteristics adapted to stationary problems, and for the nonlinear term we propose a duality method for maximal monotone operators. 35 First, we consider the variational formulation for (6): Find p E V1 = {'p E H1(L1,L ~ ) / (=P1 on x = L1, x = L2} such that + L2 (ah2*d x E L1 + ph3p*)d x 9 dx dx v'p E H&, = 0, L2). (11) Next, by using the method of characteristics for steady problems as it has already been used in lubrication problems en ', we consider the approximation: where x k ( x )= x the algorithm: - k , k being a positive small parameter.

Cm}, see Fig. , m, there exists an integer qi such that the critical point ci is qi-periodic for g p . + Then, the topological entropy of F is h ( F ) = h(f) h(gp). The previous result is based on the construction of a symbolical dynamic associated to the critical points which generates a Markov partition of the unit square 12. Two transition matrix are obtained and by the sum of the logarithms of their spectral radius the exactly topological entropy is calculated. The main idea of the program is to compute the periodic order of the critical points of the basis map f , once this is done, function gp is defined and its critical points will be computed, in the same way the periodic order of the critical points of gp has to be computed.

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